Lecture Notes on Sample Mean, Standard Deviation, and Outliers

Key Concepts

• We have data on distances between 46 retail stores and a central Distribution Center.
• The objective is to calculate the sample mean, sample standard deviation, and identify outliers in the data set.

Sample Mean Calculation

1. Data Handling:

   • Data consists of distances in miles from 46 stores to a distribution center.
   • This is a sample, not a population.

2. Process:

   • Sum up all data values.
   • Divide the sum by the number of data points (n = 46) to get the mean.
   • Store result as X̄ (sample mean).

3. Result:

   • Sample mean (X̄) = 197.2826 miles.

Sample Standard Deviation Calculation

1. Process:

   • Calculate differences between each data value and the mean (X - X̄).
   • Square each difference to eliminate direction.
   • Sum these squared differences.
   • Divide by n-1 (since it’s a sample): This is because we're using Bessel's correction.
   • Take the square root of the result to find the standard deviation.

2. Result:

   • Sample standard deviation (s) = 32.4884 miles.

Finding Outliers

1. Definition:

   • Outliers are data points that lie outside two standard deviations from the mean.

2. Bounds Calculation:

   • Upper bound = X̄ + 2s = 262.2594 miles.
   • Lower bound = X̄ - 2s = 132.3058 miles.

3. Outliers Identification:

   • Identify data values not between the bounds (132.3058 and 262.2594).

4. Results:

   • Lower end: 132 is an outlier (below the lower bound).
   • Upper end: 277 is an outlier (above the upper bound).

Conclusion

• Outliers identified are 132 and 277 miles, which deviate significantly from the average range.
• Ensure data is in order for efficient outlier identification.
• Remember to use full precision for calculations to avoid rounding errors.

Tips

• Use print function to display multiple calculations but be aware of artifact in displaying the last line twice.
• Confirm outlier values by checking both ends of the ordered data set.